Mykhaylo Shkolnikov

Limits of multilevel TASEP and similar processes

We study the asymptotic behavior of a class of stochastic dynamics on interlacing particle configurations (also known as Gelfand-Tsetlin patterns). Examples of such dynamics include, in particular, a multi-layer extension of TASEP and particle dynamics related to the shuffling algorithm for domino tilings of the Aztec diamond. We prove that the process of reflected interlacing Brownian motions introduced by Warren in \cite{W} serves as a universal scaling limit for such dynamics.

Asymptotic Analysis of Forward Performance Processes in Incomplete Markets and Their Ill-Posed HJB Equations

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g., a macroeconomic indicator) and a fast factor (e.g., stochastic volatility). We analyze the associated forward performance SPDE and provide explicit formulae for the leading order and first order correction terms for the forward investment process and the optimal feedback portfolios. They both depend on the investors initial preferences and the dynamically changing investment opportunities. The leading order terms resemble their time-monotone counterparts, but with the appropriate stochastic time changes resulting from averaging phenomena. The first order terms compile the reaction of the investor to both the changes in the market input and his recent performance. Our analysis is based on an expansion of the underlying ill-posed HJB equation, and it is supplemented b...

Small time central limit theorems for semimartingales with applications

We give conditions under which the normalized marginal distribution of a semimartingale converges to a Gaussian limit law as time tends to zero. In particular, our result is applicable to solutions of stochastic differential equations with locally bounded and continuous coefficients. The limit theorems are subsequently extended to functional central limit theorems on the process level. We present two applications of the results in the field of mathematical finance: to the pricing of at-the-money digital options with short maturities and short time implied volatility skews.

Intertwinings of beta-Dyson Brownian motions of different dimensions

We show that for all positive beta the semigroups of beta-Dyson Brownian motions of different dimensions are intertwined. The proof relates beta-Dyson Brownian motions directly to Jack symmetric polynomials and omits an approximation of the former by discrete space Markov chains, thereby disposing of the technical assumption beta>1 in [GS]. The corresponding results for beta-Dyson Ornstein-Uhlenbeck processes are also presented.